Posh recorded their first league win of the season against Reading but had gone 2-0 down when the bottle was thrown in the direction of Royals goalscorer Church.
Symns said: "I am sorry to have to report that a mindless spectator standing in the London Road end decided to throw a bottle following the scoring of the second Reading goal.
"The incident is being reviewed by the football club and the police and if the culprit is identified he will be charged and banned for life.
"This senseless action could result in a large fine or worse, closure or a points deduction.
"Do we really want these people in our stadium?"